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I transferred to School A with the intention of going there and submitted a few summer associate applications directly through the firms' websites with both my original school and School A listed on my resume, at the recommendation of School A's career office. Now, a week later, I've been accepted to School B and will be withdrawing from School A and transferring to School B instead, having never taken a class at School A and only having paid my deposit. Should I contact these firms and withdraw my applications? or send them updated materials? or do nothing? Thank you.

Just send an updated resume via e-mail to the recruiting contact on NALP and explain that you decided to transfer to School B instead of School A and that you're still interested in working there. Keep your original school on the resume though.